12.3 Multivariable Like Terms

Multivariable like terms are terms with the same variables and exponents, allowing them to be combined through addition or subtraction in algebraic expressions.

Multivariable Like Terms extends the concept of like terms to expressions containing more than one distinct variable, requiring that every variable present, along with its exponent, match exactly between two terms before they can be considered alike, regardless of the order in which the variables are written or the type of coefficient attached.

Matching an Entire Multivariable Part

Identical Multivariable Parts

Two terms with more than one variable are like terms only when their entire variable parts match completely, meaning every variable factor present in one term, together with its exponent, must also appear identically in the other term.

4 x y and - 2 x y are like terms

Reordered Variable Factors

The order in which the variable factors are written within a term does not affect whether two multivariable terms are alike, since multiplication of the variable factors is commutative, and the same product can be written in different orders while representing the identical variable part.

3 x y and 3 y x are like terms

Matching Exponents for Every Variable

Every shared variable in a multivariable term must carry the same exponent in both terms being compared; matching only some of the variables' exponents while others differ is not sufficient for the terms to be considered alike.

2 x^{2} y and 5 x^{2} y are like terms

Common Reasons Multivariable Terms Are Unlike

Missing Variable Factor

A multivariable term is not a like term to another term if it is missing a variable factor that the other term contains, even if all the variables the two terms do share have matching exponents, since a completely matching variable part requires every variable present in either term to appear in both.

3 x y and 3 x are not like terms

Different Variable Exponent Patterns

A multivariable term is not a like term to another term if the exponents attached to shared variables differ, even when both terms contain the exact same set of variable symbols, since the specific exponent pattern is as much a part of the variable part as the variables themselves.

x^{2} y and x y^{2} are not like terms

Coefficients of Multivariable Like Terms

Like Terms with Fractional Coefficients

Multivariable like terms may carry fractional coefficients, and the fractional value is treated exactly as any other coefficient, playing no role in determining whether the variable parts match, only in what value results once the terms are eventually combined.

\frac{1}{2} x y and \frac{3}{4} x y are like terms

Like Terms with Decimal Coefficients

Multivariable like terms may likewise carry decimal coefficients, again treated the same as any other coefficient for the purpose of identifying whether the terms are alike, affecting only the numerical result of any later combination.

1.5 x^{2} y and 2.5 x^{2} y are like terms

Confirming a Multivariable Match

Multivariable Like-Term Verification

Verifying that two multivariable terms are like terms means listing every distinct variable appearing in either term along with its exponent, confirming that this full list matches exactly between the two terms with no variable or exponent left unaccounted for, and only then concluding that the two terms may be combined.